While Twitter isn't completely opposed to third-party sites preserving tweets, it has its limits: the social network has effectively shut down PostGhost, a days-old service that archived the tweets of verified users with "tens of thousands" of followers. Twitter would only tell us that the site was violating its developer tool policies, such as honoring content as it's displayed on Twitter proper. However, you could also argue that it just wasn't as discriminating as Politwoops is now -- it was targeting anyone with a blue checkmark.
PostGhost maintains that it was performing a valuable service. Verified users are public figures by definition, it argues, and know that many others will see their conversations. And while Politwoops got permission for preserving political speech, PostGhost is quick to note that political statements also come from celebrities and other major figures. It's also concerned that major personalities frequently aren't held to account for their behavior, and that an archive like this offers transparency. PostGhost is "happy to continue a dialog" with Twitter to get at least some of its archive up and running once again.
The problem, as you might guess, is that Twitter won't necessarily see it that way. Not every verified user treats their account like a megaphone, after all. They might not want to chat through direct messages just to prevent tweets from resurfacing on a third-party page. If PostGhost can resume its service at all, it may have to be particularly finicky about who and what it covers.